Theorem pm5.18dc 776

Description: Relationship between an equivalence and an equivalence with some negation, for decidable propositions. Based on theorem *5.18 of [WhiteheadRussell] p. 124. Given decidability, we can consider -. (ph <-> -. ps) to represent "negated exclusive-or". (Contributed by Jim Kingdon, 4-Apr-2018.)

Assertion

Ref	Expression
pm5.18dc	|- (DECID ph -> (DECID ps -> ((ph <-> ps) <-> -. (ph <-> -. ps))))

Proof of Theorem pm5.18dc

Step	Hyp	Ref	Expression
1		df-dc 742	. 2 |- (DECID ph <-> (ph \/ -. ph))
2		pm5.501 233	. . . . . . . 8 |- (ph -> (-. ps <-> (ph <-> -. ps)))
3	2	a1d 22	. . . . . . 7 |- (ph -> (DECID ps -> (-. ps <-> (ph <-> -. ps))))
4	3	con1biddc 769	. . . . . 6 |- (ph -> (DECID ps -> (-. (ph <-> -. ps) <-> ps)))
5	4	imp 115	. . . . 5 |- ((ph /\ DECID ps) -> (-. (ph <-> -. ps) <-> ps))
6		pm5.501 233	. . . . . 6 |- (ph -> (ps <-> (ph <-> ps)))
7	6	adantr 261	. . . . 5 |- ((ph /\ DECID ps) -> (ps <-> (ph <-> ps)))
8	5, 7	bitr2d 178	. . . 4 |- ((ph /\ DECID ps) -> ((ph <-> ps) <-> -. (ph <-> -. ps)))
9	8	ex 108	. . 3 |- (ph -> (DECID ps -> ((ph <-> ps) <-> -. (ph <-> -. ps))))
10		dcn 745	. . . . . . 7 |- (DECID ps -> DECID -. ps)
11		nbn2 612	. . . . . . . . 9 |- (-. ph -> (-. -. ps <-> (ph <-> -. ps)))
12	11	a1d 22	. . . . . . . 8 |- (-. ph -> (DECID -. ps -> (-. -. ps <-> (ph <-> -. ps))))
13	12	con1biddc 769	. . . . . . 7 |- (-. ph -> (DECID -. ps -> (-. (ph <-> -. ps) <-> -. ps)))
14	10, 13	syl5 28	. . . . . 6 |- (-. ph -> (DECID ps -> (-. (ph <-> -. ps) <-> -. ps)))
15	14	imp 115	. . . . 5 |- ((-. ph /\ DECID ps) -> (-. (ph <-> -. ps) <-> -. ps))
16		nbn2 612	. . . . . 6 |- (-. ph -> (-. ps <-> (ph <-> ps)))
17	16	adantr 261	. . . . 5 |- ((-. ph /\ DECID ps) -> (-. ps <-> (ph <-> ps)))
18	15, 17	bitr2d 178	. . . 4 |- ((-. ph /\ DECID ps) -> ((ph <-> ps) <-> -. (ph <-> -. ps)))
19	18	ex 108	. . 3 |- (-. ph -> (DECID ps -> ((ph <-> ps) <-> -. (ph <-> -. ps))))
20	9, 19	jaoi 635	. 2 |- ((ph \/ -. ph) -> (DECID ps -> ((ph <-> ps) <-> -. (ph <-> -. ps))))
21	1, 20	sylbi 114	1 |- (DECID ph -> (DECID ps -> ((ph <-> ps) <-> -. (ph <-> -. ps))))

Syntax hints:  -. wn 3   -> wi 4   /\ wa 97   <-> wb 98   \/ wo 628  DECID wdc 741

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629

This theorem depends on definitions:  df-bi 110  df-dc 742

This theorem is referenced by:  xor3dc  1275  dfbi3dc  1285